Geography Reference
In-Depth Information
G=
EF
=
||D
U
x
||
.
2
+
V
2
||D
U
Y
||
2
D
U
x
(
U
)
|
Y
(
U
)
(4.13)
FG
D
U
x
(
U
)
|
Y
(
U
)
||
Y
(
U
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2
4-2 Developable Surfaces
Developable surfaces (equivalence theorem for ruled surfaces, Gauss flat surfaces, tangent
developable: developable helicoid.)
A ruled surface is called
developable
if it can be locally mapped to the plane, preserving the metric
of the surface and the generating lines. One of the lines that lies in the plane and afterwards strips
of the surface is developed on both sides of the plane, preserving both angles and lengths.
Theorem 4.2 (Equivalence theorem for ruled surfaces).
For a ruled surface, the following conditions are equivalent. (i) The surface is developable. (ii)
The surface is Gauss flat:
k
= 0. (iii) Along each of the straight lines, the surface normales are
parallel.
End of Theorem.
A ruled surface which satisfies one of the conditions (i), (ii), or (iii) is also called a
torse
or a
developable
. Every surface element without planar points which is Gauss flat (
k
= 0) is a ruled
surface.
Theorem 4.3 (Torse, developable).
An open and dense subset of every torse consists of (i) planes, (ii) cylinders, (iii) cones, and (iv)
tangent developables, namely ruled surfaces for which the vector
Y
is tangent to the directrix
x
.
End of Theorem.
A detailed proof of Theorem
4.3
is given by
Massey
(
1962
)aswellasby
Kuehnel
(
2002
, pp. 86-89).
Here, we illustrate Gauss flat surfaces of type (i) plane, (ii) cylinder, (iii) cone, and (iv) tangent
developable (“developable helicoid”) in Figs.
4.3
,
4.4
,
4.5
,and
4.6
. In contrast, Fig.
4.7
illustrates
a Gauss flat surface which is not a ruled surface based upon two segments of a cone.
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